It is important to stress that the validity of (REF ) for a dense soliton gas is far from being obvious. Indeed, as we mentioned, the very notion of the phase shift in classical soliton theory is only applicable in the context of the long-time asymptotics, i.e. when solitons have sufficient time to separate from each other after the interaction, which can only happen in a rarefied gas. In other words, the interactions between solitons can be treated as short-range only if the typical distance between solitons in a gas is much greater than soliton's width. The assumption that, in a dense gas, where solitons experience significant overlap and continual interaction, the net effect of soliton collisions on the mean velocity is expressed in the same way as in the rarefied gas requires justification.
Such a justification has been provided in [1]} in the framework of the finite-gap spectral theory [2]}.
